GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 11 Dec 2019, 20:48

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# If x and y are positive, which of the following must be

Author Message
TAGS:

### Hide Tags

Intern
Joined: 26 Mar 2013
Posts: 13
Location: India
Concentration: Finance, Strategy
Schools: Booth PT '18 (S)
Re: If x and y are positive, which of the following must be  [#permalink]

### Show Tags

17 Jan 2015, 12:26
Ans E

stat 1 & 2 were straight....but 3 though was easily simplified....took smtym to confirm

Good tricky qtn
CEO
Joined: 20 Mar 2014
Posts: 2560
Concentration: Finance, Strategy
Schools: Kellogg '18 (M)
GMAT 1: 750 Q49 V44
GPA: 3.7
WE: Engineering (Aerospace and Defense)
If x and y are positive, which of the following must be  [#permalink]

### Show Tags

28 Jan 2016, 10:41
noboru wrote:
If x and y are positive, which of the following must be greater than $$\frac{1}{\sqrt{x+y}}$$?

I. $$\frac{\sqrt{x+y}}{2}$$
II. $$\frac{\sqrt{x}+\sqrt{y}}{2}$$
III. $$\frac{\sqrt{x}-\sqrt{y}}{x+y}$$

A. I only
B. II only
C. III only
D. I and II only
E. None

In "MUST BE TRUE" questions, you need to use POE and be ready to prove that none of the options/statements are possible. Even 1 not possible scenario will make that option not allowed.

Had this question been "could be true" instead, II only would have been correct with x=y=1. But as this is a MUST BE TRUE question, you need to make sure to find 1 set of (x,y) that will negate the given conditions.

i) and iii) can be easily POE-d by assuming x=y=1. In both these cases the resulting values will NOT BE GREATER than $$1/(x+y)^{0.5}$$.

For ii), you can clearly see x=y=1 gives you a value greater than $$1/(x+y)^{0.5}$$ but what about x=y=4 again you get a value greater. So lets take x=y=0.25. In this case you will end up getting ii) < $$1/(x+y)^{0.5}$$. Hence this expression as well is NOT always true and is hence eliminated.

As you eliminated all the 3 possible options, OA must be E (none).

Hope this helps.
Intern
Joined: 30 Sep 2016
Posts: 7
Re: If x and y are positive, which of the following must be  [#permalink]

### Show Tags

29 Nov 2016, 07:23
Hi.

Would you mind explaining why I got the wrong answer? I assumed x as 3 and y as 6 as we know they are positive integers. Is it wrong for me to assume this?
Math Expert
Joined: 02 Sep 2009
Posts: 59674
Re: If x and y are positive, which of the following must be  [#permalink]

### Show Tags

29 Nov 2016, 07:43
Nasahtahir wrote:
Hi.

Would you mind explaining why I got the wrong answer? I assumed x as 3 and y as 6 as we know they are positive integers. Is it wrong for me to assume this?

The question asks which of the following MUST be greater than ... So, which is ALWAYS greater than ... If it's greater for some particular set of numbers it does not mean that it will be greater for other sets.
_________________
Board of Directors
Status: QA & VA Forum Moderator
Joined: 11 Jun 2011
Posts: 4834
Location: India
GPA: 3.5
Re: If x and y are positive, which of the following must be  [#permalink]

### Show Tags

29 Nov 2016, 09:13
noboru wrote:
If x and y are positive, which of the following must be greater than $$\frac{1}{\sqrt{x+y}}$$?

I. $$\frac{\sqrt{x+y}}{2}$$
II. $$\frac{\sqrt{x}+\sqrt{y}}{2}$$
III. $$\frac{\sqrt{x}-\sqrt{y}}{x+y}$$

A. I only
B. II only
C. III only
D. I and II only
E. None

Plug in an try -

$$x = 9$$ & $$y = 16$$

$$\frac{1}{\sqrt{x+y}}$$ = $$\frac{1}{25}$$ $$= 0.04$$

I. $$\frac{\sqrt{x+y}}{2}$$ $$= \frac{5}{2} =2.5$$

II. $$\frac{\sqrt{x}+\sqrt{y}}{2}$$ $$= \frac{3 + 4}{2}$$ $$= 3.5$$

Option III is a bit different , there can be 2 cases -

$$x = 9$$ & $$y = 16$$ & $$x = 16$$ & $$y = 9$$

III. $$\frac{\sqrt{x}-\sqrt{y}}{x+y}$$

Thus there can be multiple possible solutions for option (III)

Hence, we are confident about I & II, but option III , may or may not be > $$\frac{1}{\sqrt{x+y}}$$ , so answer will be (E) None of the above.
_________________
Thanks and Regards

Abhishek....

PLEASE FOLLOW THE RULES FOR POSTING IN QA AND VA FORUM AND USE SEARCH FUNCTION BEFORE POSTING NEW QUESTIONS

How to use Search Function in GMAT Club | Rules for Posting in QA forum | Writing Mathematical Formulas |Rules for Posting in VA forum | Request Expert's Reply ( VA Forum Only )
GMAT Club Legend
Joined: 12 Sep 2015
Posts: 4134
Re: If x and y are positive, which of the following must be  [#permalink]

### Show Tags

25 Oct 2017, 07:13
Top Contributor
noboru wrote:
If x and y are positive, which of the following must be greater than $$\frac{1}{\sqrt{x+y}}$$?

I. $$\frac{\sqrt{x+y}}{2}$$
II. $$\frac{\sqrt{x}+\sqrt{y}}{2}$$
III. $$\frac{\sqrt{x}-\sqrt{y}}{x+y}$$

A. I only
B. II only
C. III only
D. I and II only
E. None

Let's test some values.

x = 1 and y = 1
1/√(x + y) = 1/√(1 + 1) = 1/√2

I. √(x + y)/2 = √(1 + 1)/2 = √2/2
Notice that, if we take 1/√2 and multiply top and bottom by √2, we get: √2/2, which is the same as quantity I
Since quantity I is not greater than 1/√2, statement I is not true

II. (√x + √y)/2 = (√1 + √1)/2 = (1 + 1)/2 = 2/2 = 1
Since 1 IS greater than 1/√2, we cannot say for certain whether quantity II will always be greater than √(x + y)/2

III. (√x - √y)/(x + y) = (√1 - √1)/(1 + 1) = (1 - 1)/2 = 0/2 = 0
Since 0 is not greater than 1/√2, statement III is not true

So, statements I and III are definitely not true, and we aren't yet 100% certain about statement II
Let's try another pair of values for x and y

x = 0.25 and y = 0.25
1/√(x + y) = 1/√(0.25 + 0.25) = 1/√0.5
Let's further simplify 1/√0.5
Since 1 = √1, we can say: √1/√0.5
Then we'll use a rule that says (√k)/(√j) = √(k/j)
So, √1/√0.5 = √(1/0.5) = √2
We see that, when x = 0.25 and y = 0.25, 1/√(x + y) = √2

II. (√x + √y)/2 = (√0.25 + √0.25)/2 = (0.5 + 0.5)/2 = 1/2
Since 1/2 is NOT greater than √2, statement II is not true

_________________
Test confidently with gmatprepnow.com
Non-Human User
Joined: 09 Sep 2013
Posts: 13744
Re: If x and y are positive, which of the following must be  [#permalink]

### Show Tags

14 Nov 2019, 23:53
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
Re: If x and y are positive, which of the following must be   [#permalink] 14 Nov 2019, 23:53

Go to page   Previous    1   2   3   [ 47 posts ]

Display posts from previous: Sort by